# MA232A - Euclidean and non-Euclidean Geometry Michaelmas Term 2015 Dr. David R. Wilkins Resources related to Gauss's General Investigations of Curved Surfaces

## Resources related to Gauss's General Investigations of Curved Surfaces

“The single most important work in the history of differential geometry is Gauss' paper of 1827, Disquisitiones generales circa superficies curvas.”

M. Spivak, A comprehensive introduction to differential geometry (2nd edn, 1979), vol. 2, p. 74.

### The Text of Gauss's General Investigations of Curved Surfaces

The text of Gauss's paper is available, in the translation into English by J.C. Morehead and A.M. Hildebeitel, and also in the original Latin, at the following URLS:—

Carl Friedrich Gauss, General Investigations of Curved Surfaces, translated by James Cadall Morehead and Adam Miller Hildebeitel, in English, Project Gutenberg, 1902)
Carl Friedrich Gauss, Disquisitiones generales circa superficies curvas (1828, Internet Archive, in Latin)

### Notes on the Text of Gauss's General Investigations

Gauss's General Investigations: The Differential Geometry of Curved Surfaces. [N.B., These notes are work in progress and are thus draft, provisional, incomplete and subject to change.]

### Vector Algebra and Spherical Geometry

Section~2 to Gauss's General Investigations of Curved Surfaces contains some results concerning spherical trigonometry. Gauss proved these using the Cosine Rule for Spherical Trigonometry, a result that can be obtained without much difficulty using coordinate geometry. Gauss's results can also be obtained as fairly direct applications of the algebra of vectors in three-dimensional Euclidean space. The following material is available:---

Notes on Vector Algebra and Spherical Trigonometry
This begins with an account of basic vector algebra, defining and describing scalar and vector products in three-dimensional vector algebra, relating these products to lengths of vectors and angles between them. Standard identities of vector algebra are derived, including the basic properties of the scalar triple product, the Vector Triple Product Identity and Lagrange's Quadruple Product Identity. These results are then applied to obtain basic identities of spherical trigonometry that are used or proved in Section 2 of Gauss's General Investigations of Curved Surfaces

### Properties of Smooth Surfaces

The following notes are based on the contemporary approach to the theory of smooth surfaces and their tangent spaces, using theorems of real analysis that were developed in the century following Gauss's publication of the Disquisitiones generales circa superficies curvas in 1828:

Notes on Smooth Surfaces
These notes begin with a summary of the definition and basic properties of differentiability and smoothness for functions of several real variables. This is followed by a discussion of smooth curvilinear coordinate systems over open sets in three-dimensional Euclidean space. The definition of smooth surfaces is introduced, and followed by a discussion of smooth local coordinates on a smooth surface. The tangent space to a smooth surface at a point of that surface is then defined and its basic properties are discussed. The notes then explain how differentials of smooth functions can be viewed as linear functionals on the tangent spaces of smooth surface. Finally some applications of the Inverse Function Theorem of real analysis in several variables with particular relevance to the theory of smooth surfaces are developed.
Curvature of Smooth Surfaces in Three-Dimensional Space
These notes develop the theory of the Gauss Map developed in Sections 4 onwards of Gauss's General Investigations of Curved Surfaces.

### The Hyperbolic Plane

The Hyperbolic Plane
These notes develop the theory of conformally-flat Riemannian metrics on open subsets of the plane, and applies this theory in the study of the Hyperbolic Plane.

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